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Percolation, by G. Grimmett, Geoffrey Grimmett
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Percolation theory is the study of an idealized random medium in two or more dimensions. The emphasis of this book is upon core mathematical material and the presentation of the shortest and most accessible proofs. Much new material appears in this second edition including dynamic and static renormalization, strict inequalities between critical points, a sketch of the lace expansion, and several essays on related fields and applications.
- Sales Rank: #3332137 in Books
- Published on: 1989-05-01
- Original language: English
- Number of items: 1
- Dimensions: 9.75" h x 6.50" w x .75" l, .0 pounds
- Binding: Hardcover
- 320 pages
From the Back Cover
Percolation theory is the study of an idealized random medium in two or more dimensions. It is a cornerstone of the theory of spatial stochastic processes with applications in such fields as statistical physics, epidemiology, and the spread of populations. Percolation plays a pivotal role in studying more complex systems exhibiting phase transition. The mathematical theory is mature, but continues to give rise to problems of special beauty and difficulty. The emphasis of this book is upon core mathematical material and the presentation of the shortest and most accessible proofs. The book is intended for graduate students and researchers in probability and mathematical physics. Almost no specialist knowledge is assumed beyond undergraduate analysis and probability. This new volume differs substantially from the first edition through the inclusion of much new material, including: the rigorous theory of dynamic and static renormalization; a sketch of the lace expansion and mean field theory; the uniqueness of the infinite cluster; strict inequalities between critical probabilities; several essays on related fields and applications; numerous other results of significant. There is a summary of the hypotheses of conformal invariance. A principal feature of the process is the phase transition. The subcritical and supercritical phases are studied in detail. There is a guide for mathematicians to the physical theory of scaling and critical exponents, together with selected material describing the current state of the rigorous theory. To derive a rigorous theory of the phase transition remains an outstanding and beautiful problem of mathematics.
Most helpful customer reviews
8 of 8 people found the following review helpful.
Percolation
By james lawry
Grimmett's book, Percolation, is excellent.
Percolation theory began in the 50's; its mathematics is now quite mature, but the theory has recently acquired new techniques because many of the questions initially raised by percolation theory are still unanswered.
Percolation technology is now a cornerstone of the theory of disordered systems, and the methods of this book are now being extended into dynamical systems theory and the life sciences. This book covers the mathematics of percolation theory, presenting the shortest rigorous proofs of the main facts. Many problems in percolation theory are beautiful, but some of the apparent simplicity of the subject is deceiving, because the subject is quite deep. Grimmett cuts through many of the difficulties presenting the important concepts clearly and sucinctly.
The author restricts himself- for accessibility to the maximum readership-to bond percolation on a cubic lattice. Grimmett presents the core material at a graduate level for folks conversant with elementary probability theory and real analysis. Having some knowledge of ergodic theory, graph theory, and some mathematical physics helps, however. There is litle discussion of continuous, mixed, inhomogenous, long range, first passage or oriented percolation.
Beginning with existance of Psubc for the edge probability p we arrive at an infinite open cluster followed by discusssion of the basic techniques of the FKC, BK inequalities and Russo's formula. Grimmett then discusses open clusters per vertex and subcritical percolation, beginning with the Aizeman-Barsky and Menshikov methods for identifying the critical point, followed by a systematic study of the subcritical phase. He then discusses supercritical percolation, including 2 dimensional percolation, continuum percolation and random processes. The author gives a full list of references.
3 of 3 people found the following review helpful.
Excellent
By A Customer
The latest edition of Dr Grimmett's Percolation is surely the best book on the subject. He presents topics as clearly as possible without neglecting the technical details. His writing style is very readable, making much of this book accessible even to those who don't have all the necessary background in mathematics to understand all the proofs. Anyone looking for an easy introduction to the topic would be better off with Stauffer's book. But to gain any moderate understanding of this fascinating subject, and the methods and results of current research, this is the only book to have.
0 of 0 people found the following review helpful.
Really Good!!
By PST
The book treats percolation mainly on the d-dimensional lattice, only rarely are other graphs considered.
The treatment is excellent in my opinion, Prof. Grimmett knows the subject!
The special mathematical prerequisites seem moderate, some knowledge of probabiliy theory and very little knowledge of grpah theory as sufficient.
The proofs of the theorems are generally such that they are easy to follow. Prof. Grimmett simply adds a line or two, where other authors make larger leaps; I prefer Prof. Grimmett's style!
Last but not least, the book is virtually typo-free! I found only two easy ones in the whole book. This is very helpful, if the book is used for self-study (as I did)
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